082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

2.3 Basic concepts of functions 63
Example2.6 If f (x) = 5x verifythat the inverse of f isgiven by f −1 (x) = x 5 .
Solution The function f receives an input of x, and produces an output of 5x. Hence when the
inverse function, f −1 , receives an input of 5x, itproduces anoutput ofx, thatis
f −1 (5x) =x
We introduce a new variable,z, given by
so
z=5x
Then
x = z 5
f −1 (z)=x= z 5
Writing f −1 withxas the argument gives
f −1 (x) = x 5
Example2.7 If f (x) = 2x +1,find f −1 (x).
Solution The function f receives an input of x and produces an output of 2x + 1. So when the
inverse function, f −1 , receives aninputof2x +1 itproducesanoutput ofx, thatis
f −1 (2x+1)=x
We introduce a new variable,z, defined by
z=2x+1
Rearranginggives
So
x = z −1
2
f −1 (z)=x= z −1
2
Writing f −1 withxas the argument gives
f −1 (x) = x −1
2